Advanced calculus
Free
Description
Contents
Reviews
Language
English
ISBN
0-86720-122-3
Title Page
Preface
Contents
Chapter 0: Introduction
0.1 Logic: Quantifiers
0.2 The Logical Connectives
0.3 Negations of Quantifiers
0.4 Sets
0.5 Restricted Variables
0.6 Ordered Pairs and Relations
0.7 Functions and Mappings
0.8 Product Sets; Index Notation
0.9 Composition
0.10 Duality
0.11 The Boolean Operations
0.12 Partitions and Equivalence Relations
Chapter 1: Vector Spaces
1.1 Fundamental Notions
1.2 Vector Spaces and Geometry
1.3 Product Spaces and Hom(V,W)
1.4 Affine Subspaces and Quotient Spaces
1.5 Direct Sums
1.6 Bilinearity
Chapter 2: Finite-Dimensional Vector Spaces
2.1 Bases
2.2 Dimension
2.3 The Dual Space
2.4 Matrices
2.5 Trace and Determinant
2.6 Matrix Computations
2.7* The Diagonalization of a Quadratic Form
Chapter 3: The Differential Calculus
3.1 Review in R
3.2 Norms
3.3 Continuity
3.4 Equivalent Norms
3.5 Infinitesimals
3.6 The Differential
3.7 Directional Derivatives; the Mean-Value Theorem
3.8 The Differential and Product Spaces
3.9 The Differential and R^n
3.10 Elementary Applications
3.11 The Implicit-Function Theorem
3.12 Submanifolds and Lagrange Multipliers
3.13* Functional Dependence
3.14* Uniform Continuity and Function-Valued Mappings
3.15* The Calculus of Variations
3.16* The Second Differential and the Classification of Critical Points
3.17* Higher Order Differentials. The Taylor Formula
Chapter 4: Compactness and Completeness
4.1 Metric Spaces; Open and Closed Sets
4.2* Topology
4.3 Sequential Convergence
4.4 Sequential Compactness
4.5 Compactness and Uniformity
4.6 Equicontinuity
4.7 Completeness
4.8 A First Look at Banach Algebras
4.9 The Contraction Mapping Fixed-Point Theorem
4.10 The Integral of a Parameterized Arc
4.11 The Complex Number System
4.12* Weak Methods
Chapter 5: Scalar Product Spaces
5.1 Scalar Products
5.2 Orthogonal Projection
5.3 Self-Adjoint Transformations
5.4 Orthogonal Transformations
5.5 Compact Transformations
Chapter 6: Differential Equations
6.1 The Fundamental Theorem
6.2 Differentiable Dependence on Parameters
6.3 The Linear Equation
6.4 The nth-Order Linear Equation
6.5 Solving the Inhomogeneous Equation
6.6 The Boundary-Value Problem
6.7 Fourier Series
Chapter 7: Multilinear Functionals
7.1 Bilinear Functionals
7.2 Multilinear Functionals
7.3 Permutations
7.4 The Sign of a Permutation
7.5 The Subspace A^n of Alternating Tensors
7.6 The Determinant
7.7 The Exterior Algebra
7.8 Exterior Powers of Scalar Product Spaces
7.9 The Star Operator
Chapter 8: Integration
8.1 Introduction
8.2 Axioms
8.3 Rectangles and Paved Sets
8.4 The Minimal Theory
8.5 The Minimal Theory (Continued)
8.6 Contented Sets
8.7 When is a Set Contented?
8.8 Behavior Under Linear Distortions
8.9 Axioms for Integration
8.10 Integration of Contented Functions
8.11 The Change of Variables Formula
8.12 Successive Integration
8.13 Absolutely Integrable Functions
8.14 Problem Set: The Fourier Transform
Chapter 9: Differentiable Manifolds
9.1 Atlases
9.2 Functions, Convergence
9.3 Differentiable Manifolds
9.4 The Tangent Space
9.5 Flows and Vector Fields
9.6 Lie Derivatives
9.7 Linear Differential Forms
9.8 Computations with Coordinates
9.9 Riemann Metrics
Chapter 10: The Integral Calculus on Manifolds
10.1 Compactness
10.2 Partitions of Unity
10.3 Densities
10.4 Volume Density of a Riemann Metric
10.5 Pullback and Lie Derivatives of Densities
10.6 The Divergence Theorem
10.7 More Complicated Domains
Chapter 11: Exterior Calculus
11.1 Exterior Differential Forms
11.2 Oriented Manifolds and the Integration of Exterior Differential Forms
11.3 The Operator d
11.4 Stokes' Theorem
11.5 Some Illustrations of Stokes' Theorem
11.6 The Lie Derivative of a Differential Form
Appendix I: "Vector Analysis"
Appendix II: Elementary Differential Geometry of Surfaces in E^3
Chapter 12: Potential Theory in E^n
12.1 Solid Angle
12.2 Green's Formulas
12.3 The Maximum Principle
12.4 Green's Functions
12.5 The Poisson Integral Formula
12.6 Consquences of the Poisson Integral Formula
12.7 Harnack's Theorem
12.8 Subharmonic Functions
12.9 Dirichlet's Problem
12.10 Behavior Near the Boundary
12.11 Dirchlet's Principle
12.12 Physical Applications
12.13 Problem Set: The Calculus of Residues
Chapter 13: Classical Mechanics
13.1 The Tangent and Cotangent Bundles
13.2 Equations of Variation
13.3 The Fundamental Linear Differential Form on T*(M)
13.4 The Fundamental Exterior Two-Form on T*(M)
13.5 Hamiltonian Mechanics
13.6 The Central-Force Problem
13.7 The Two-Body Problem
13.8 Lagrange's Equations
13.9 Variational Principles
13.10 Geodesic Coordinates
13.11 Euler's Equations
13.12 Rigid Body Motion
13.13 Small Oscillations
13.14 Small Osccillations (Continued)
13.15 Canonical Transformations
Selected References
Notation Index
Index
The book hasn't received reviews yet.
